8 TOPO LOGICAL GROUPS 33 (If 0 is such that 0 < 0 < 1, then ||/(X) -/(x)|| £ \\f(6x') -f(8x)\\ - 2M(1 - 0), and we can choose 0 so that d(6x, E — U) > i Vk2-l \\x' — x\\, f(ex)\\^me\\X'-X\\ and m6\\X'-x\\ -2M(1 - 8. TOPOLOGICAL GROUPS If G is a group, in which the law of composition is written (for example) multiplicatively, a topology on G is said to be compatible with the group structure if the two mappings (x, y)t~* xy of G x G (endowed with the product topology) into G, and x^x"1 of G into G, are continuous. A group endowed with a topology compatible with its group structure is called a topological group. We leave it to the reader to transcribe this definition (and all the results which follow) into additive notation. An isomorphism of a topological group G onto a topological group G' is by definition an isomorphism of the group G onto the group G' which is bicon- tinuous. If G' = G, we say automorphism instead of isomorphism. If G is any group, the law of composition (x9y)t-+yx defines a group structure on the set G (and this structure is different from the given one unless G is commutative). The group so defined is denoted by G° and is called the opposite of G. Any topology which is compatible with the group structure of G is also compatible with that of G°, and hence makes G° into a topological group. The mapping xt-^-x'1 is then an isomorphism of G onto G°. Examples (12.8.1) The discrete topology and the chaotic topology (12.1.1) are com- patible with the structure of any group. The topology of a normed space (in particular R or C) is compatible with its additive group structure. On the additive group Q of rational numbers, the topology defined by the /?-adic distance d (3.2.6) is compatible with the group structure, because by virtue of the definition of this distance and (3.2.6.4) we have d(x0 +yQ,x + y)£ Max(rf(*0 , x), d(y0 , y)) and If E is a (real or complex) Banach space and GL(E) the set of linear homeo- morphisms of E onto E, then the topology induced on GL(E) by that of ; E) is compatible with the group structure ((5 J.5) and (8.3.2)).ed in U. If L is the closed